TITLE: Are there non-homeomorphic 3-manifolds with the same Turaev-Viro-Barrett-Westbury invariants?
QUESTION [7 upvotes]: The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but independent of the choice of) a triangulation of $M$ and $\mathcal{C}$-data labelings.
See the book Turaev-Virelizier (2017) on Section 13.1, and the papers Turaev-Viro (1991) and Barrett-Westbury (1996).
Question: Are there non-homeomorphic closed oriented $3$-manifolds $M$ and $N$ such that $|M|_{\mathcal{C}} = |N|_{\mathcal{C}}$, for all spherical fusion category $\mathcal{C}$?
This answer by Meng Cheng provides a formula for $|M|_{\mathcal{C}}$ with $M=L(p,q)$ a Lens space and $\mathcal{C}=Vec_{G,\omega}$ a pointed fusion category, and shows that it distinguishes $L(7,1)$ from $L(7,2)$, for a good choice of $G$ and $\omega$.

REPLY [9 votes]: I asked Alexis Virelizier (coauthor of the book mentioned above) by e-mail. Here is his answer (reproduced with his authorisation):

The answer is yes. See Theorem 1.1 (page 2291) in the following paper
by Funar: https://www-fourier.ujf-grenoble.fr/~funar/2013geomtopol.pdf